Factorize each of the following expressions:
(1) `16(a-b)^3-24(a-b)^2`
(2) `a b(x^2+1)+x(a^2+b^2)`
(3) `a^2x^2+(a x^2+1)x+a`
(4) `a(a-2b-c)+2b c`

Explanation

(i)`16(a-b)^3-24(a-b)^2`
`=2 xx (a-b) xx 8(a-b)^2-3 xx 8(a-b)^2`
`=8(a-b)(a-b)(2a-2b-3)`

(ii)`a b(x^2+1)+x(a^2+b^2)`
`=abx^2+aby^2−a^2xy−b^2xy`
`= abx^2 –b^2xy+aby^2−a^2xy`
`= bx(ax−by)−ay(ax−by)`
`= (ax−by)(bx−ay)`

(iii)`a^2x^2+(a x^2+1)x+a`
`=ax^2 +(a^2-1)x+a`
`=ax^2+a^2x-x-a`

(iv) `a(a-2b-c)+2b c`
`= a2–2ab–ac+2bc`
`= a (a – 2b) – c (a – 2b)`
`= (a – 2b) (a – c)`